**Answer:**

86.65% probability that the mean monitor life would be greater than 74.4 months in a sample of 85 monitors

**Step-by-step explanation:**

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

**Normal probability distribution**

When the distribution is normal, we use the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

**Central Limit Theorem**

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean and standard deviation , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean and standard deviation .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

**In this question:**

Population:

Sample of 85:

**If the claim is true, what is the probability that the mean monitor life would be greater than 74.4 months in a sample of 85 monitors**

This is 1 subtracted by the pvalue of Z when X = 74.4. So

By the Central Limit Theorem

has a pvalue of 0.1335

1 - 0.1335 = 0.8665

86.65% probability that the mean monitor life would be greater than 74.4 months in a sample of 85 monitors